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Simulation of optical near and far fields of dielectric apertureless scanning probes
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2006 Nanotechnology 17 475
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INSTITUTE OF PHYSICS PUBLISHING
NANOTECHNOLOGY
Nanotechnology 17 (2006) 475–482
doi:10.1088/0957-4484/17/2/022
Simulation of optical near and far fields of
dielectric apertureless scanning probes
R Esteban, R Vogelgesang and K Kern
Max Planck Institute für Festkörperforschung, Stuttgart, Germany
Received 18 May 2005, in final form 15 November 2005
Published 21 December 2005
Online at stacks.iop.org/Nano/17/475
Abstract
We study apertureless field enhancing optical probes beyond the spherical
approximation in a smooth transition towards up to 3 µm long conical
silicon tips. Such tips are used in apertureless scanning near field optical
microscopy, which holds the promise of sub 10 nm lateral resolution. A
fully three-dimensional numerical solution to the Maxwell equations is
obtained with the multiple multipole method giving simultaneously both
near fields and scattered far fields. The significance of focused beam
excitation for work with long tips is illustrated and the relative influence of
relevant length scales such as tip length, excitation wavelength, and beam
waist radius is discussed. In the limit of vanishing tip apex radius, the near
field grows without bounds, whereas the far field remains finite. We verify
that for small apex radii the near field confinement at the tip apex, which is
related to the achievable lateral resolution, scales with the inverse of the
radius. We find, however, that long tips exhibit a markedly lower
confinement than spherical or very short tips. Relevant for experimental
studies, we demonstrate how scanning the excitation field with long conical
tips can be a useful technique for mapping the focal volume. We show that
the normalized near field at the tip apex is robustly tolerant against small
misalignments or misorientations of illumination focus and tip apex.
1. Introduction
One of the most promising optical imaging techniques
for achieving lateral resolution of a few nanometres is
the scattering type apertureless scanning near field optical
microscope (aSNOM) [1, 2]. It utilizes localized, strong
near field enhancement at the apex of a scanning probe
tip, belonging typically to a scanning tunnelling microscope
(STM) or an atomic force microscope (AFM). The local
optical response of a sample can be determined with a lateral
resolution essentially given by the effective apex radius.
Images based on elastic scattering have been demonstrated at
frequencies ranging from the visible to the microwave [3–5].
Inelastic spectroscopic phenomena such as Raman scattering
and photoluminescence are also accessible [6, 7].
There are two limiting modes of operation of an aSNOM,
namely, as a passive re-scatterer of local optical fields into the
far field [8] and as an active source whose own localized near
field excites an optical response of exclusively those sample
volumes that are in close proximity to the tip apex [6]. In
either case, as the probe tip scans a sample surface, changes in
0957-4484/06/020475+08$30.00 © 2006 IOP Publishing Ltd
the far field can be observed which are related with the optical
properties of the sample volume just below the tip, as well as
local topography, and instrumental parameters [9–11].
In general, neither the passive nor the active aSNOM mode
is realized in its pure form, because the optical responses of the
probe tip and sample surface also interact with each other and
require one to consider the tip–sample system as a whole. If the
localized field enhancement at the tip apex acting as an active
source is strong enough to overwhelm the parasitic scattering
from the surrounding area and the bulk of the tip that enters the
far field collection optics, a direct recording of the local optical
properties is possible. Otherwise, a sophisticated amplification
and discrimination scheme has to be employed [12–15].
A pivotal assumption for field enhancing near field optical
microscopy is that the observable far field is directly related
(proportional in the simplest case) with the fields at the apex
of the field enhancing tip. In other words, the bulk of the tip
may effectively be disregarded. Correspondingly, this nearto-far field transition is frequently taken to be appropriately
modelled by representing the whole tip as a dipole polarizable
sphere centred in the spherical apex cap featuring a scalar or
Printed in the UK
475
R Esteban et al
tensorial polarizability [16, 17]. Spectroscopic trends which
enter through the wavelength dependence of the dielectric
constant were found to be adequately captured by these
simple models. Simulating a spherical tip can give important
additional insights [18] but it was soon found that the actual tip
shape may indeed have considerable impact on the near field
enhancement and observable far field intensities [19–23].
In experimental aSNOM work one frequently uses tips
which are metal-coated or even fully metallic. Their negative
real part of the dielectric constant can lead to polariton
resonances which increase the attainable signal considerably.
Especially for nanoscopic structures, however, these are very
sensitively coupled with the exact geometry and relative
orientation of tip and sample through the phenomenon of quasistatic resonances [23]. In this paper, we wish to concentrate
on those geometrical aspects that are decoupled from such
resonances. Therefore, we choose a positive real part dielectric
material like silicon at visible frequencies. Our second
motivation for simulating Si tips concerns the achievable
spatial resolution of aSNOM instruments, which is related to
the tip apex radius. In this context, Si tips are a prime candidate
for ultimate resolution well below 10 nm thanks to processing
technology readily available for semiconductors but not for
metals. Preliminary experimental results demonstrated to us
the feasibility of using Si tips.
For the principal geometry, which is closely related to
our own experimental set-up, refer to figure 1. First, conical
tips of a wide range of apex radius and length are studied—
up to 3 µm in length—which establishes relevant model
parameters for realistic elongated tips to facilitate subsequent
studies of full tip–sample models (to be published separately).
Afterwards, for long enough tips, the parameters defining
the incident radiation beam are studied, namely, the angle of
incidence, the Gaussian beam waist radius and the position of
the tip relative to the optical focus. The results obtained are
significant for optimizing experimental set-ups and exemplify
how certain model simplifications, when overdone, may lead
to discrepancies with respect to experimental results.
We stress the significance even of isolated tips as
representative models for the passive mode aSNOM. In
conjunction with the spatially variable field of a focused
beam excitation we determine the relation of local fields and
observable far field quantities. This allows us to verify the
fundamental assumption for passive mode aSNOM that the
scattered far field intensity scales with the local excitation field
at the tip’s apex [8]. It also suggests a means to experimentally
characterize the incident beam and align the tip in the proper
focal position.
2. Simulated model
In our model, we study the interaction of a focused Gaussian
illumination beam with conical tips of up to 3 µm length,
whose hemispherical apex has a radius as small as 2 nm. Thus
it belongs to a class of multi-scale electrodynamic models
that presents a number of challenges [24]. An elegant and
numerically not too demanding solution is obtained with the
generalized multipole technique (GMT) [25–27].
The GMT divides space into domains of different material
properties, with the interfaces separating the domains as
476
(a)
(b)
ε material
L
R
1m
y
E
k
Detector
θ
ε vacuum
x
z
Figure 1. Geometry of the model tips considered and example of
field patterns obtained. (a) Conical Si tips of length L and apex
radius R are illuminated in vacuum by a Gaussian beam at an angle
θ with respect to the tip axis. The scattered far field is recorded in
backscattering direction. (b) Typical simulation result exhibiting the
near fields outside a long tip and a standing wave pattern inside with
a characteristic length scale ∼λmatter . A logarithmic scale is used
because linear contrast would allow us only to observe the strongest
fields near the apex. The isolines represent an increase of ×1.3 in
the field modulus.
boundaries.
It uses a set of expansion series of the
electromagnetic field, for example multipoles or planar waves,
which are perfect solutions to the three-dimensional Maxwell
equations. The field in the domain is represented by one or
more such expansion series, which are parameterized by an
appropriate number of coefficients. These coefficients are
the free parameters of the model that need to be determined,
for example, by least-squares minimization of the error in
the boundary conditions at the discretized domain interfaces.
These technical aspects of the modelling are largely automated
in the numerical platform MAX-1 that we use [28]. Still, the
location and type of the different expansion series must be
selected with care to avoid interdependences.
The geometry of the simulated tip and illumination as
well as collection optics is shown in figure 1(a). Unless stated
differently in the text, the following parameters apply. The tips
are of conical shape, with a half angle of 10◦ . The upper and
the lower ends of the cone are capped by two hemispheres,
forming a convex C 1 continuous surface. The radius of the
smaller cap (the tip apex) is 10 nm. The tip material is silicon
(r = 17.76 + 0.508i [29]), embedded in vacuum.
The illumination is a Gaussian beam of λvac = 514 nm,
corresponding to one of the principal wavelengths of an Ar +
ion laser. The implementation in the program corresponds to
a fifth-order correction of the paraxial Gaussian beam [30–32]
and is accurate near the focus for not too tight beams. The
beam is in-plane linearly polarized with an angle of incidence
θ with respect to the axis of the tip of 70◦ and a waist radius
500 nm. To illustrate the simulations, figure 1(b) presents the
distribution of the electric field for a 1.4 µm long tip.
In the different results presented, the excitation radiation
is suppressed and only the values of the scattered near and
far fields are considered, as they are the ones of interest in
aSNOM. The near field enhancement is defined as the ratio
of the maximum average field modulus of the excited field
Simulation of optical near and far fields of dielectric apertureless scanning probes
Figure 2. Study of the convergence of the scattered far field when
the number of parameters is increased, at tips of different lengths.
(evaluated near the apex) to the maximum field modulus of
the incident beam. The back-scattered far field is obtained by
integrating the average Poynting vector in a circular detector
covering a solid angle of NA = 0.17, centred in the trajectory
of the backscattered beam (figure 1). The simulated distance
to the tip is one metre, to guarantee that only the far field is
measured.
We apply two simultaneous criteria for acceptable
convergence of the results. First, we accept only results with an
average relative error along the boundary less than 0.5%. The
error must not be significantly affected when other boundary
points are evaluated than those used in the error minimization.
Second, in each data set assembled to one graph we checked
for a significant number of models the effect of increasing
the total number of free parameters in the field expansions by
∼50%. The variation in the obtained near field enhancement
and scattered far field intensity must be less than 1%. We
took special care in the limiting cases of the model considered.
Figure 2 serves to illustrate the good convergence of the results,
by showing the evolution of the scattered far field with the
number of parameters for typical tips of lengths between 1.3
and 1.4 µm. In this particular example, our two criteria are
first verified by the curve with 1232 free parameters.
3. Results
3.1. Influence of tip shape
In figure 3(a), the influence of the length L of the tip on the
near field enhancement and scattered far field is shown for the
case of Gaussian illumination. Three regions of qualitatively
different behaviour can be identified, namely a monotonic
increase for lengths less than λvac /2, an oscillatory behaviour
between λvac /2 and 2 µm, and an approximately constant
behaviour beyond that.
Starting from a length of 20 nm, i.e., a sphere of 10 nm
radius, the near field enhancement increases almost linearly
with the length L due to a mostly constructive contribution
originating in all the material, up to L ≈ 200 nm. We found
that in this tip length range the scattered far field intensity scales
approximately with the volume V in a power law with exponent
2.55, which contrasts with the usual expectation derived from
Rayleigh scattering theory, predicting a V 2 behaviour for small
particles [33]. This we traced to the shape being changed in
the transition from an elongated conical shape (L = 200 nm)
to a spherical tip (L = 20 nm). If instead the tip is scaled
uniformly, preserving the conical shape, we obtain the familiar
V 2 law.
Beyond L ≈ 200 nm resonances appear. They are
related to the length scales given by the wavelengths outside
and inside the tip on the one hand and tip size parameters
on the other—notably the tip length and, as long as it is
strongly illuminated, the radius of the top hemispherical cap.
The first maximum corresponds to a length slightly bigger
than λvac /2, i.e., the first resonance value expected for linear
antennas [34]. Such behaviour has already been observed
in the equivalent two-dimensional case [35]. As the tip gets
bigger, the resonances become less pronounced, though they
are still non-negligible even at lengths of around 1 µm. In this
region, the resonance length λmatter /2 appears more important,
in correspondence with the standing wave patterns that form
as illustrated in figure 1(b). The Fourier transform of the near
field enhancement and scattered far fields for tips between 1
and 1.8 µm (not shown here) indicates a high spectral density
close to λmatter /2. This is especially clear in the simulation of
artificial tips of real and positive r , which show no damping
of internal waves.
At tip lengths of around six times the illumination beam
waist radius, the hemispherical top cap is not efficiently
illuminated anymore and the waves generated near the tip are
damped away before they are reflected at the top. The results
converge to an almost constant level. Between 2.8 and 3 µm,
the variation is less than ±4%.
In figure 3(b) the same results are shown as in (a) except
for the use of planar wave excitation. In this case the whole
tip is always fully illuminated, including the hemispherical
top cap. For small lengths, the results are almost identical to
those obtained with focused Gaussian illumination. However,
for bigger lengths the use of planar illumination affects the
results dramatically, making it very difficult to extrapolate to
an ‘infinite tip’. The change is especially accentuated for the
scattered far field, as the influence of the hemispherical top cap
dominates the results.
The second tip shape parameter we studied is the radius
of its apex. Realistic probe tips feature effective radii of
significantly less than 100 nm, good ones less than 10 nm.
Figure 4 shows the evolution of the near field enhancement and
the scattered far field intensity with the apex radius in this range
of interest. The length of the tip is kept constant at 3 µm, where
the influence of length-related resonances is nearly negligible.
For radii below ∼30 nm the symmetric point exactly below the
tip apex is also the location of the strongest near field. Above
∼30 nm, however, two values of enhancement—the actual
maximum and the value just below the tip—are shown because
the location of the field maximum is shifted significantly away
from the exact apex of the tip due to retardation effects. As
expected, the near field enhancement diverges towards zero
radius. The expression 13.2–2.83 ln(R/nm) describes the near
field enhancement quite accurately for radii R between 2 and
30 nm. The scattered far field also increases for smaller radii,
although it does not diverge but it saturates.
Not only are the scattered intensity and the maximum of
the near field modulus of interest but also the local distribution
and field lines of the near field. Figure 5(a) presents the
477
R Esteban et al
(b)
(a)
Figure 3. Near field enhancement and scattered far field when the length of the tip is modified, for Gaussian beam (a), and planar wave (b)
illumination.
where E 0 and Y0 are fitting parameters. Y0 is of the order of R
and we have found it to depend only weakly on the illumination
properties (the beam radius and others). Moreover, we
observed an excellent rotational symmetry of the solution in
the horizontal x z plane below the apex which suggests further
generalization to a cylindrically invariant expression [36].
3.2. Influence of the illumination
Figure 4. Evolution of the near and far field for apex radius varying
between 2 and 60 nm.
strength of the field E along horizontal lines below the apex
of the tip. In the vicinity of the apex, it is predominantly
vertically oriented—along the y axis. Notice the reduction
of the confinement when the vertical distance from the tip is
increased (figure 5(b)). At a 5 nm distance, a full width at
half maximum of ∼32.5 nm is obtained, to be compared with
∼18.5 nm just below the apex. As the confinement of the field
is related to the resolution achievable when scanning a sample,
it is favorable to use a very small tip–substrate distance. For
tips longer than a few hundred nanometres the shape of the apex
fields is essentially independent of length. Very small tips,
however, produce more confined fields than longer tips, both
in the horizontal and the vertical direction [27]. For example,
the full width at half maximum just below the tip is ∼13.5 nm
for the sphere. Evidently, simulating small or even spherical
tips may overestimate the resolution by several tens of per cent.
The normalized fields scale spatially quite accurately with
the radius, as shown in figure 5, where the shapes of the field in
a horizontal line at R/2 below the tip are plotted, with R being
the radius of the tip. The vertical distance is chosen as R/2
to illustrate how the results scale with R not only horizontally
but also vertically, for distances of the order of R.
We note that the results converge surprisingly well to a
Lorentzian shape for tips of a few hundred nanometres length
or more. They are proportional to
|E(x)| ∝ E 0 +
478
1
Y02 + x 2
(1)
In this subsection, the influence of the illumination used is
studied. When considering not just a planar wave excitation,
but Gaussian illumination, besides the angle of incidence
and polarization, the Gaussian beam waist radius and the
relative location of tip apex and focus centre are relevant
parameters. Qualitatively, the consequences can be understood
as collective retardation effects, with different parts of the
whole tip volume oscillating in phase. Depending on the
details of how a specific illumination beam excites the different
partial volumes, their overall interference may be constructive
or destructive. Sensitive dependence of this interference on
beam waist radius and angle of incidence θ of the beam can be
expected.
Figure 6 shows the effect of variable beam waist radius,
keeping constant the amplitude at the focus. For very tight
illumination, under the standard conditions defined above
(θ = 70◦ ), the near field enhancement and scattered far field
are maximal. Destructive overall interference from excitation
of a larger tip volume reduces the fields for growing radii [37].
As can be seen from figure 6, whereas the near field at
the apex continues to fall for increasing radii, the far field
intensity exhibits a local minimum around 700 nm radius and
is dominated eventually by the hemispherical region at the top
of the tip. As expected, in the limit of infinite beam waist
radius both the near and far field values converge to the planar
wave illumination case.
For θ = 20◦ (not shown here), we find the trend quite
altered in the sense of a more constructive excitation of larger
tip volumes. From a minimum in near and far field scattering
for tight illumination both intensities increase monotonically
towards their respective plane-wave excitation values.
Figure 7 shows the influence of changing θ on the field
strengths for a 3 µm long tip. The near field enhancement
is minimal for θ = 0◦ and 180◦ when the linear polarization
vector of the electrical field is perpendicular to the axis of
Simulation of optical near and far fields of dielectric apertureless scanning probes
(a)
(b)
Figure 5. (a) Near field modulus (normalized to the maximum) along a horizontal cut at R/2 below the apex for different radii. (b) Near
field modulus (relative to the field modulus at the focus of the incident radiation) for R = 10 nm along horizontal lines located at different
distances below the tip apex.
Figure 6. Near field enhancement and scattered far field intensity
for different radii of the Gaussian excitation beam, asymptotically
approaching those of a planar wave excitation for increasing beam
waist radius.
the tip [36, 38]. The maximum is however not observed
at 90◦ (where the electrical field vector is exactly parallel
to the axis of the tip) but at a different angle [39], here
θ ∼ 40◦ . The scattered far field intensity presents four
maxima. One corresponds to the near field enhancement
maximum at ∼40◦ and two to the direct illumination of the top
hemispherical cap—already seen to be strongly radiating, but
of little relevance to experimental measurements. The fourth
maximum, located at ∼100◦ , corresponds to illumination
perpendicular to the side surface of the cone. The behaviour
of the normalized modulus of the electric fields near the apex
is approximately constant for θ = 30◦ . . . 180◦ .
The near field enhancement and far field intensity when
the beam focus is not centred at the apex but displaced in the
y direction is presented in figure 8. As long as the top cap of
the tip is much less illuminated than the apex of the tip (here
y waist radius), both values scale quite closely with the local
field of the excitation at the apex of the tip—the square of the
field modulus in the case of the scattered far field intensity.
This suggests an appealing way of characterizing the beam
used in experimental set-ups [17]. Equivalent conclusions
are obtained when displacing the illumination beam in the z
and x directions. The normalized near field modulus near the
apex remains approximately unchanged, as long as the tip apex
remains in the focal volume given by the waist radius and the
Figure 7. Evolution of the results as a function of the angle θ
between the beam propagation direction and the axis of a 3 µm tip.
Near field enhancement and scattered far field intensity. A local
maximum around θ ∼ 40◦ is found.
Figure 8. Near field enhancement and scattered far field intensity
for displacements of the focus along the y axis. Zero displacement
corresponds to the location of the apex of the tip. Displacements
towards positive values correspond to illumination of the upper part
of the tip. The secondary maxima at ∼1.25 µm are related to the
direct illumination of the top cap. For comparison, we plot with a
dotted line the appropriately scaled modulus of the excitation field at
the apex of the tip with the near field enhancement and its square
with the scattered far field intensity for the corresponding
displacement of the focus.
Rayleigh length. Thus, small misalignments will likely not
critically affect the images obtained when scanning a sample.
479
R Esteban et al
4. Discussion
The near field enhancement obtained at the apex of sufficiently
long Si tips upon illumination with focused 514 nm radiation
settles at a value around 7. This is in the low range of
field enhancements reported in related studies, which go from
values less than 10 up to more than 70 [35, 36, 40, 41].
The difference derives partly from the different materials and
shapes of the tips, but mostly from the presence or absence of
geometrical and plasmon or other polariton resonances. The
present study concentrates on the effects of tip geometry and
the consequences of variations in the alignment of the exciting
radiation. For Si at visible frequencies the real part of the
dielectric constant is large and positive, exhibiting no polariton
resonances that would affect the results.
Different length scales are relevant to this study. Only for
sufficiently long tips and only for focused illumination do we
find near field enhancement and far field scattering intensity to
reach values that are stable against further increases of the tip
length. That is, the ratio of beam focus and tip length plays an
important role. A planar wave excitation can be inappropriate
in a simulation when the scattered far field intensity is desired,
as the results are likely dominated by the bulk or the upper
termination of the tip and give little information about the
fields at the apex of the tip. From additional studies we find
that the dampening effect of the imaginary part of the index of
refraction—that is, the ratio of penetration depth to tip length—
appears to be equally important. A small but finite extinction
coefficient ensures that waves generated near the apex do not
travel too far inside the domain and form extended standing
wave patterns. Otherwise these would lead to an increased
sensitivity of the near and far fields to changes in tip geometry
and illumination parameters as is characteristic of undamped
resonance phenomena.
The simulation of very short tips or the extreme case
of spherical tips leads to somewhat misleading results as
they significantly underestimate the far field intensity and
may overestimate the achievable resolution. The greater field
confinement of short tips can be analysed by considering the
radial distance dependence of the near field intensity, which is
critical in sample scanning, where the tip–substrate interaction
depends sensitively on the distance r . The electric near
field of a simple dipole polarizable sphere scales as r −3 . In
contrast, equation (1) suggests an r −2 behaviour relative to
an appropriately chosen origin inside the conical tips with
spherical apex studied here. Comparing our result to the
near fields at the point tip of perfect infinite cones, which are
known to exhibit an r ν−1 behaviour with the critical exponent
ν > 0 depending on the cone angle as well as the dielectric
material [42], we believe the Lorentzian behaviour in our
case is coincidental. For other cone angles, we expect the
exponent to differ from −2, which will be the subject of
further studies. Nevertheless, the different distance scaling
behaviours of the near fields for spherical and long tips could
particularly affect the dynamic variants of scattering-type near
field optical methods, which periodically modulate the probe–
sample distance.
The observed antenna-like resonance at a tip length of
∼λvac /2 verifies the possibility of specifically designed superresonant field enhancement structures [20]. Their realization,
480
however, requires a precise manufacturing process and the
excess near field enhancement of only some 60% over the
long-tip value makes the effort questionable for this particular
structure. Furthermore, the increase in the observable far field
scattered intensity (more than four times the value of very
long tips) results from all of the small tip’s volume, and the
contribution from only the local near field at the apex might be
even more difficult to extract experimentally than in the case
of long tips.
Over a wide range of apex radii, the field confinement
scales inversely proportionally to the radius. Reducing the
radius will likely result in a proportional improvement in
resolution. Moreover, the scattered far field increases with
reducing radius, which suggests that highly sharpened tips are
also desirable for optimizing the received signal. The far field
increase, however, remains finite and does not diverge as the
near field enhancement, for reasons of energy conservation,
which allows infinite near fields in an infinitesimal volume,
but requires finite cross sections for the scattered far fields
(figure 4).
The consideration of Gaussian beams in addition to planar
wave excitations opens the possibility of maximizing near
field enhancement and scattered far field by varying the
characteristics of the excitation. For example, the finding of an
optimal angle of incidence (figure 7), which differs from the
case of polarization exactly along the tip axis, is relevant to the
implementation of near-field optical microscopes, especially
for opaque samples, which have to be illuminated under similar
conditions. However, it is necessary to keep in mind that the
presence of a substrate will in general modulate the optimal
angle due to the exciting radiation resulting from interference
of the incident beam with the waves reflected by the substrate
surface.
Including the substrate is a substantial step for further
studies. Nonetheless, we expect many of our present results
to carry over to such simulations. Examples are: the need
of simulating confined illumination for long tips, significant
antenna resonances for tip lengths less than at least a few µm,
the considerable increase of near field enhancement and field
confinement for decreasing apex radius, and the presence of a
maximum for illumination polarization not parallel to the tip
axis.
An important aspect of the present paper is the study of
off-apex illumination, because it serves as a test of how near
field properties can be related to those recorded in the far field
in the passive aSNOM mode. Figure 8 illustrates how, under
certain circumstances, a faithful near-to-far field transition is
indeed facilitated by the use of near field enhancing tips. The
experimentally accessible scattered far field intensity scales
quite accurately with the square of the local field modulus of
the excitation fields at the apex. This holds in our tips as
long as the top cap is not strongly excited in comparison with
the volume near the apex. With respect to the essential nearto-far field scattering relation being proportional, the simple
spherical models are thus validated.
Two caveats must be noted, however. First, the gradient
of the local excitation fields used in the present study is small
on the length scale of the tip apex radius, i.e., |∂/∂x| 1/R. At this point it is still an open question how well
the intensity a far field detector registers corresponds to the
Simulation of optical near and far fields of dielectric apertureless scanning probes
local fields at a sharp tip probing near fields that vary with
equally high spatial frequency. Second, as already mentioned,
when the tip approaches a substrate it exhibits both active and
passive behaviour. Considering the tip–substrate interaction
considerably complicates the study. In this scenario, the
relation between the local excitation field and the scattered
far field intensity is likely no longer a simple proportionality.
Finally, we emphasize the immediate applications of our
results to the experimental work. As long as the upper part of
the tip is not overilluminated, the simple relationship between
excitation fields, local near fields, and scattered far fields
allows one to map the three-dimensional focus by scanning
the tip apex through the focal volume. That is, this method
constitutes a convenient method for three-dimensional beam
focus characterization. Also, the maximum found when
the excitation is focused at the apex greatly simplifies the
alignment of experimental set-ups like that of figure 1, an
approach successfully employed in the alignment of our own
experimental set-up. Further, it can be observed that slight
off-focus alignment of less than the focal waist radius or small
variations of the angle of incidence of a Gaussian beam exciting
the apex of long tips do not affect the near fields at the apex
too much. Apart from a uniform scaling factor, the spatial
field structure is rather tolerant against somewhat sub-optimal
alignment of the excitation.
5. Conclusion
In this paper we studied how both the scattered near and far
fields of optically excited conical tips evolve with tip length and
apex radius, with the angle of incidence and the waist radius
of the Gaussian excitation beam, as well as with the relative
alignment of tip apex and beam focus.
The near field enhancement and scattered far field intensity
vary considerably when tips are considered that are short
compared to the beam waist radius and the penetration depth
but level out for long enough tips. Accordingly, appropriately
long tips should be used together with sufficiently focused
excitation beams to obtain steady behaviour for both the near
field enhancement at the tip apex and the recorded far fields.
The simulation of too short or spherical tips also results in
more confined fields, which can lead to overestimating the
achievable lateral resolution as well as a misjudgment of the
distance dependence of the near field mediated tip–sample
interaction.
Reducing the apex radius gives rise to a proportional
increase in near field confinement, which is expected to
translate to better lateral resolution. Our simultaneous
modelling of near and far field scattering demonstrates how
for vanishing radius the near field enhancement grows without
bounds. But the detectable scattered far field intensity, that
is, the experimentally relevant quantity, remains finite in
accordance with energy conservation.
We found an optimal angle of incidence, θ ∼ 40◦ , for
maximal near field strength as well as scattered far field
intensity. Note that this does not realize an electric field
vector parallel to the isolated tip’s axis. Different tip materials
or geometries, of the presence of a sample surface, whose
reflections also excite the tip, will likely alter the optimal angle,
requiring further numerical studies.
Finally, the direct correspondence between scattered far
field intensity and the square modulus of the incident field at
the apex suggests a convenient method for three-dimensional
focus characterization as well as alignment of tip apex and
optical focus. The near field spatial distribution around the
tip apex is also rather tolerant against small misalignments
or misorientations, with only a uniform reduction in field
strength. This suggests that field enhancing near field optical
microscopy may indeed be developed into a robust tool for
routine investigations of optical properties at the nanometre
scale.
References
[1] Zenhausern F, Martin Y and Wickramasinghe H K 1995
Science 269 1083
[2] Knoll B and Keilmann F 1999 Nature 399 134
[3] Knoll B, Keilmann F, Kramer A and Guckenberger R 1997
Appl. Phys. Lett. 70 2667
[4] Keilmann F, Knoll B and Kramer A 1999 Phys. Status Solidi b
215 849
[5] Taubner T, Hillenbrand R and Keilmann F 2003 J. Microsc.
210 311
[6] Hartschuh A, Sanchez E J, Xie X S and Novotny L 2003 Phys.
Rev. Lett. 90 095503
[7] Hu D, Micic M, Klymyshyn N, Suh Y D and Lu H P 2003 Rev.
Sci. Instrum. 74 3347
[8] Hillenbrand R, Keilmann F, Hanarp P, Sutherland D S and
Aizpurua J 2003 Appl. Phys. Lett. 83 368
[9] Madrazo A, Nieto-Vesperinas M and Garcı́a N 1996 Phys. Rev.
B 53 3654
[10] Carminati R and Greffet J-J 1995 J. Opt. Soc. Am. A 12 2716
[11] Martin O J F, Girard C and Dereux A 1996 J. Opt. Soc. Am. A
13 1801
[12] Hillenbrand R and Keilmann F 2000 Phys. Rev. Lett. 85 3029
[13] Maghelli N, Labardi M, Patanè S, Irrera F and
Allegrini M 2001 J. Microsc. 202 84
[14] Walford J N, Porto J A, Carminati R, Greffet J-J, Adam P M,
Hudlet S, Bijeon J-L, Stashkevich A and Royer P 2001
J. Appl. Phys. 89 5159
[15] Fikri R, Barchiesi D, H’Dhili F, Bachelot R, Vial A and
Royer P 2003 Opt. Commun. 221 13
[16] Koglin J, Fischer U C and Fuchs H 1997 Phys. Rev. B 55 7977
[17] Bouhelier A, Beversluis M R and Novotny L 2003 Appl. Phys.
Lett. 82 4596
[18] Porto J A, Johansson P, Apell S P and López-Rı́os T 2003
Phys. Rev. B 67 085409
[19] Kottmann J P, Martin O J F, Smith D R and Schultz S 2001
Phys. Rev. B 64 235402
[20] Krug J T II, Sánchez E J and Xie X S 2002 J. Chem. Phys.
116 10895
[21] Calander N and Willander M 2002 J. Appl. Phys. 92 4878
[22] Renger S, Grafström J and Eng L 2005 Phys. Rev. B
71 075410
[23] Fredkin D R and Mayergoyz I D 2003 Phys. Rev. Lett.
91 253902
[24] Girard C and Dereux A 1996 Rep. Prog. Phys. 59 657
[25] Hafner C 1999 Post-Modern Electromagnetics: Using
Intelligent Maxwell Solvers (Chichester: Wiley)
[26] Moreno E, Erni D, Hafner C and Vahldieck R 2002 J. Opt.
Soc. Am. A 19 101
[27] Bohn J L, Nesbitt D J and Gallagher A 2001 J. Opt. Soc. Am.
A 18 2998
[28] Hafner C 1998 Max-1: A Visual Electromagnetics Platform
(Chichester: Wiley)
[29] Aspnes D E and Studna A A 1983 Phys. Rev. B 27 985
[30] Davis L W 1979 Phys. Rev. A 19 1177
481
R Esteban et al
[31] Barton J P and Alexander D R 1989 J. Appl. Phys. 66 2800
[32] Evers T, Dahl H and Wriedt T 1996 Electron. Lett. 32 1356
[33] Van de Hulst H C 1981 Light Scattering by Small Particles
(New York: Dover)
[34] Collin R E and Zucker F J 1969 Antenna Theory (New York:
McGraw-Hill)
[35] Crozier K B, Sundaramurthy A, Kino G S and Quate C F 2003
J. Appl. Phys. 94 4632
[36] Novotny L, Bian R X and Xie X S 1997 Phys. Rev. Lett.
79 645
482
[37] Martin Y C and Wickramasinghe H K 2002 J. Appl. Phys.
91 3363
[38] Martin O J F and Girard C 1997 Appl. Phys. Lett. 70 705
[39] Sun W X and Shen Z X 2003 J. Opt. Soc. Am. A 20 2254
[40] Martin Y C, Hamann H F and Wickramasinghe H K 2001
J. Appl. Phys. 89 5774
[41] Micic M, Klymyshyn N, Suh Y D and Lu H P 2003 J. Phys.
Chem. B 107 1574
[42] Van Bladel J 1991 Singular Electromagnetic Fields and
Sources (Oxford: Oxford University Press)
IOP PUBLISHING
NANOTECHNOLOGY
Nanotechnology 19 (2008) 169801 (1pp)
doi:10.1088/0957-4484/19/16/169801
Corrigendum
Simulation of optical near and far fields of dielectric
apertureless scanning probes
R Esteban, R Vogelgesang and K Kern Nanotechnology
17 475–482
The value given in line four of the first paragraph of page three
for the numerical aperture used to collect the scattered radiation
should read ‘NA=0.342’ instead of ‘NA=0.17’. The paper is
otherwise unaffected.
0957-4484/08/169801+01$30.00
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